Additive properties of central Drazin invertibility of elements in a ring

Abstract

For two central Drazin invertible elements $a$ and $b$ of a ring, we first prove that $a+b$ is central Drazin invertible under the condition of $ab=0$. Then we establish the relation between central Drazin invertibility of $a+b$ and $1+a^cb$ under some generalized commutative condition, of which the explicit representations of central Drazin inverses are given. When $a$ and $b$ are two central group invertible elements, additive properties of central group inverses are studied under the conditions of $ab=ba$ and $abb^{\copyright}=baa^{\copyright}$, respectively.